Andrew Jackson Essay Example | Topics and Well Written Essays - 750 words - 1

Andrew Jackson - Essay Example On his defense of the rights of the Federal government during the Nullification Crisis of 1832, Jackson invoked his presidential powers to affirm the superiority of the federal constitution. 1Jackson declares that the responsibility 'imposed on him by the constitution" is to ensure that the laws are applied 'faithfully' as long as the execution of his duties is consistent with the authority 'emphatically pronounced in the constitution'. This contention delineates further Jackson's respect of the supremacy of the federal law which bestowed on him the authority to apply it objectively and fairly, making sure that he remains loyal to his obligations within legal bounds. Jackson is also unswerving on his perspective towards the role of the federal government as an overseer of the states. For him, South Carolina's illusory power to nullify a federal law is 'inconsistent with every principle on which [the union] was founded." Jackson's unwavering fidelity to fulfill his obligations as president and to put into practice the fed... This does not connote constraints on the freedom of the states but rather assert the duties of the federal government towards problems that fall beyond the bounds of the states' jurisdictions. 3Without this control, states' would effect its own policy towards the Indians which could result to more complications. Such policy makes it possible for the states and the national government to avoid collision. 4The prevention of this collision which purports to 'preserve the Union by all constitutional means" delineates Jackson's faith in federalism and the ideals he bestows in his office. 5This only shows that Jackson is ready to apply the full force of the law in order to protect the federal principles which for him form a part of a 'happy union.' In his response towards South Carolina's Nullification issue, he proclaims that he will 'recourse to force' to preserve the Union and views further opposition towards the national government's promulgation of the federal law a form of treason. However, this does not try to pit the state authorities against the national government but intends to fortify the federal institutions which all states subjected themselves to. As there had been other states in the past who aimed to weaken the federal institution by threatening to break away with the union, Jacks on's assertion of the power of the central government to bring to a halt these threats resulted to the strengthening of the fundamental system and avoided confrontations between the national government and the states. Jackson's decision not to renew the contract of the Bank of the United States further attested to his consistency in his obligations to the people

Sugary Drinks or Diet Drinks Essay Example for Free

Sugary Drinks or Diet Drinks Essay Abstract Better beverage choices can help fight and prevent obesity and diabetes. Water, of course, is the best beverage option. It delivers everything the body needs—pure H2O—with zero calories. But for some tastes, plain water is just too plain—and it may be unrealistic to ask everyone to kick the sugar-water habit overnight. We must instead work to retrain the American palate away from sweet drinks. Cutting our taste for sweetness will require concerted action on several levels—from creative food scientists and marketers in the beverage industry, as well as from individual consumers and families, schools and worksites, and state and federal government. Sugary Drinks or Diet Drinks? What’ the Best Choice? Soft drinks are the beverage of choice for millions of Americans. Some drink them morning, noon, night, and in between. They’re tasty, available everywhere, and inexpensive. They’re also a prime source of extra calories that can contribute to weight gain. Once thought of as innocent refreshment, soft drinks are also coming under scrutiny for their contributions to the development of type 2 diabetes, heart disease, and other chronic conditions. Diet soft drinks, made with artificial sweeteners, may not be the best alternatives to regular soft drinks. The term â€Å"soft drink† covers a lot of ground. It refers to any beverage with added sugar or other sweetener, and includes soda, fruit punch, lemonade and other â€Å"ades,† sweetened powdered drinks, and sports and energy drinks. In this section of The Nutrition Source, we focus on non-alcoholic sweetened drinks. Drunk every now and then, these beverages wouldn’t raise an eyebrow among most nutrition experts, any more than does the occasional candy bar or bowl of ice cream. But few people see them as treats. Instead, we drink rivers of the stuff. According to figures from the beverage industry, soft drink makers produce a staggering 10. 4 billion gallons of sugary soda pop each year. That’s enough to serve every American a 12-ounce can every day, 365 days a year. The average can of sugar-sweetened soda or fruit punch provides about 150 calories, almost all of them from sugar, usually high-fructose corn syrup. That’s the equivalent of 10 teaspoons of table sugar (sucrose). If you were to drink just one can of a sugar-sweetened soft drink every day, and not cut back on calories elsewhere, you could gain up to 15 pounds in a year. Soft Drinks and Weight Historians may someday call the period between the early 1980s and 2009 the fattening of America. Between 1985 and now, the proportion of Americans who are overweight or obese has ballooned from 45 percent in the mid-1960s to 66 percent today. (The Centers for Disease Control and Prevention has an online slide show that shows the spread of obesity in the U.S. ) There’s no single cause for this increase; instead, there are many contributors. One of them is almost certainly our penchant for quenching our thirst with beverages other than water. Once upon a time, humans got almost all of their calories from what nature put into food. That changed with the advent of cheap sugar, and then cheaper high-fructose corn syrup. High-fructose corn syrup has been fingered as one of the villains in the obesity epidemic, but in fact, table sugar and corn sweeteners likely have the same physiological impact on blood sugar, insulin, and metabolism. Sugar added to food now accounts for nearly 16 percent of the average American’s daily intake. Sweetened soft drinks make up nearly half of that. Dozens of studies have explored possible links between soft drinks and weight. It isn’t an easy task, for several reasons (read Sorting Out Studies on Soft Drinks and Weight to learn why). Despite these research challenges, studies consistently show that increased consumption of soft drinks is associated with increased energy intake. In a meta-analysis of 30 studies in this area, 10 of 12 cross-sectional studies, five of five longitudinal studies, and four of four long-term experimental studies showed this positive association. A different meta-analysis of 88 studies showed that the effect appeared to be stronger in women, studies focusing on sugar-sweetened soft drinks, and studies not funded by the food industry: Studies in children and adults have also shown that cutting back on sugary drinks can lead to weight loss. On the surface, it makes sense that the more ounces of sugar-rich soft drink a person has each day, the more calories he or she takes in. Yet that runs counter to what happens with solid foods. People tend to compensate for a bigger than usual meal or for a snack by taking in fewer calories later. That’s how weight stays stable. This compensation doesn’t seem to happen with soft drinks. No one knows for sure why this happens, but there are several possibilities: Fluids may not be as satiating as solid foods. That means they don’t provide the same feeling of fullness or satisfaction that solid foods do, which might prompt you to keep eating. The body doesn’t seem to â€Å"register† fluid calories as carefully as it does those from solid food. This would mean they are added on top of calories from the rest of the diet. It is possible that sweet-tasting soft drinks—regardless of whether they are sweetened with sugar or a calorie-free sugar substitute—might stimulate the appetite for other sweet, high-carbohydrate foods. Use headings and subheadings to organize the sections of your paper. The first heading level is formatted with initial caps and is centered on the page. Do not start a new page for each heading. Subheading Subheadings are formatted with italics and are aligned flush left. Citations Source material must be documented in the body of the paper by citing the authors and dates of the sources. The full source citation will appear in the list of references that follows the body of the paper. When the names of the authors of a source are part of the formal structure of the sentence, the year of the publication appears in parenthesis following the identification of the authors, for example, Smith (2001). When the authors of a source are not part of the formal structure of the sentence, both the authors and years of publication appear in parentheses, separated by semicolons, for example (Smith and Jones, 2001; Anderson, Charles, Johnson, 2003). When a source that has three, four, or five authors is cited, all authors are included the first time the source is cited. When that source is cited again, the first author’s surname and â€Å"et al. † are used. See the example in the following paragraph. Use of this standard APA style â€Å"will result in a favorable impression on your instructor† (Smith, 2001). This was affirmed again in 2003 by Professor Anderson (Anderson, Charles Johnson, 2003). When a source that has two authors is cited, both authors are cited every time. If there are six or more authors to be cited, use the first author’s surname and â€Å"et al. † the first and each subsequent time it is cited. When a direct quotation is used, always include the author, year, and page number as part of the citation. A quotation of fewer than 40 words should be enclosed in double quotation marks and should be incorporated into the formal structure of the sentence. A longer quote of 40 or more words should appear (without quotes) in block format with each line indented five spaces from the left margin. 1 References Anderson, Charles Johnson (2003). The impressive psychology paper. Chicago: Lucerne Publishing. Smith, M. (2001). Writing a successful paper. The Trey Research Monthly, 53, 149-150. Entries are organized alphabetically by surnames of first authors and are formatted with a hanging indent. Most reference entries have three components: Authors: Authors are listed in the same order as specified in the source, using surnames and initials. Commas separate all authors. When there are seven or more authors, list the first six and then use â€Å"et al. † for remaining authors. If no author is identified, the title of the document begins the reference. Year of Publication: In parenthesis following authors, with a period following the closing parenthesis. If no publication date is identified, use â€Å"n. d. † in parenthesis following the authors. Source Reference: Includes title, journal, volume, pages (for journal article) or title, city of publication, publisher (for book). Appendix Each Appendix appears on its own page. Footnotes 1Complete APA style formatting information may be found in the Publication Manual. Table 1 Type the table text here in italics; start a new page for each table [Insert table here] Figure Captions Figure 1. Caption of figure [Figures – note that this page does not have the manuscript header and page number].

Deir El Mdina Essay -- essays research papers

Deir El Medina Describe the village of Deir El Medina. The village of Deir El Medina grew from the time of the 18th Dynasty to the 20th. By its final stage approximately 70 houses stood within the village walls and 50 outside. Perhaps 600 people lived here by then. A wall surrounded the village approximately six meters high built of mud-brick. Gates were located at each end. The villages of Deir El Medina made up a special government department under the vizier of Upper Egypt, and were a select largely hereditary group of scribes, quarrymen, stonemasons, artisans, and labourers, who created the final resting place for their divine rulers. Describe in detail a typical workers house at Deir El Medina. Most of the houses in Deir El Medina were built in a standard elongated design, 15 by 5 meters. They had rubble bases and mud brick superstructures, and shared walls like today’s terrace housing. Each of these houses would have the following features. Down several steps from the street was an entrance room, with niches for offerings, stalae and busts. Often there were painted images, sometimes of the god Bes. A low bed-like structure has suggested to some archaeologists that the entrance room was also used as a birthing room. A doorway led into the main room of the house, with raised dais by one wall, plastered and whitewashed. Against another wall may have been a small altar and offering table and niches for household gods. A small cellar was often located under this room, approached by a small flight of steps and covered by a wooden trapdoor. Several small rooms may have led off the main room, possibly for sleeping, work or storage. At the rear was a small walled court, which served as the kitchen. It contained an oven for baking bread, a small grain storage silo, a container for water and grinding equipment. Another family shrine and another small cellar may also have been here. A staircase led to the roof where the family might sleep or store goods. Windows were normally set high in the walls with a grill. Though the outside of the houses was whitewashed, traces of paintings have been found in the interior walls. Refer to diagram 1.1 What type of furniture existed in such a household? The furniture was generally well made and often beautifully crafted. Nobles’ furniture was often inlaid with semi-precious stones and ivory and the villages often copied ... ... and grape juice were commonly consumed by workers- wines were more expensive. Spices and herbs were used such as cinnamon, cumin and thyme. ENTERTAINMENT- There is abundant information about leisure pursuits of Egyptian nobility. They hunted wild game such as the ibex, ostriches, gazelles, hares and wildfowl, and fished in the Nile. It is not certain if the villagers shared these pursuits. Villagers enjoyed music from instruments such as the harp, lyre, lute, flute and drum. Board games such as senet were also popular. What was Egyptian Family life like? Houses held five to six people yet burials often included at least three generations. Marriages were generally arranged. There was no ceremony but complex legal arrangements were made. Divorce was simple; reasons given range from adultery to infertility or simple apathy. Women had considerable legal, economic and social status. Some even appeared to be literate. Children played like they do in every culture yet are often shown performing light work. Boys were educated in a nearby temple where they were taught reading, writing and arithmetic. Squabbles between families, and even within families appear to have been quite common. Deir El Mdina Essay -- essays research papers Deir El Medina Describe the village of Deir El Medina. The village of Deir El Medina grew from the time of the 18th Dynasty to the 20th. By its final stage approximately 70 houses stood within the village walls and 50 outside. Perhaps 600 people lived here by then. A wall surrounded the village approximately six meters high built of mud-brick. Gates were located at each end. The villages of Deir El Medina made up a special government department under the vizier of Upper Egypt, and were a select largely hereditary group of scribes, quarrymen, stonemasons, artisans, and labourers, who created the final resting place for their divine rulers. Describe in detail a typical workers house at Deir El Medina. Most of the houses in Deir El Medina were built in a standard elongated design, 15 by 5 meters. They had rubble bases and mud brick superstructures, and shared walls like today’s terrace housing. Each of these houses would have the following features. Down several steps from the street was an entrance room, with niches for offerings, stalae and busts. Often there were painted images, sometimes of the god Bes. A low bed-like structure has suggested to some archaeologists that the entrance room was also used as a birthing room. A doorway led into the main room of the house, with raised dais by one wall, plastered and whitewashed. Against another wall may have been a small altar and offering table and niches for household gods. A small cellar was often located under this room, approached by a small flight of steps and covered by a wooden trapdoor. Several small rooms may have led off the main room, possibly for sleeping, work or storage. At the rear was a small walled court, which served as the kitchen. It contained an oven for baking bread, a small grain storage silo, a container for water and grinding equipment. Another family shrine and another small cellar may also have been here. A staircase led to the roof where the family might sleep or store goods. Windows were normally set high in the walls with a grill. Though the outside of the houses was whitewashed, traces of paintings have been found in the interior walls. Refer to diagram 1.1 What type of furniture existed in such a household? The furniture was generally well made and often beautifully crafted. Nobles’ furniture was often inlaid with semi-precious stones and ivory and the villages often copied ... ... and grape juice were commonly consumed by workers- wines were more expensive. Spices and herbs were used such as cinnamon, cumin and thyme. ENTERTAINMENT- There is abundant information about leisure pursuits of Egyptian nobility. They hunted wild game such as the ibex, ostriches, gazelles, hares and wildfowl, and fished in the Nile. It is not certain if the villagers shared these pursuits. Villagers enjoyed music from instruments such as the harp, lyre, lute, flute and drum. Board games such as senet were also popular. What was Egyptian Family life like? Houses held five to six people yet burials often included at least three generations. Marriages were generally arranged. There was no ceremony but complex legal arrangements were made. Divorce was simple; reasons given range from adultery to infertility or simple apathy. Women had considerable legal, economic and social status. Some even appeared to be literate. Children played like they do in every culture yet are often shown performing light work. Boys were educated in a nearby temple where they were taught reading, writing and arithmetic. Squabbles between families, and even within families appear to have been quite common.

Leadership Styles :: essays research papers

The style of leadership that would be the most comfortable for me would be delegation. One of the first signs of good supervision is effective delegation. Delegation is when supervisors give responsibility and authority to subordinates to complete a task, and let the subordinates figure out how the task can be accomplished. Effective delegation develops people who are ultimately more fulfilled and productive. Managers become more fulfilled and productive themselves as they learn to count on their staffs and are freed up to attend to more strategic issues. Delegating is a critical skill for supervisors. Delegating involves working with an employee to establish goals, granting them sufficient authority and responsibility to achieve the goals, often giving them substantial freedom in deciding how the goals will be achieved, remaining available as a resource to help them achieve the goals, assessing their performance, addressing performance issues and/or rewarding their performance. Ultimately, the supervisor retains responsibility for the attainment of the goals, but chooses to achieve the goals by delegating to someone else. True delegation means giving up a little of what we would like to hold onto while keeping what we might prefer to give up. Participative style of leadership would be least comfortable for me. A participative leader, rather than making decisions, looks to involve other staff in the process. Often however, as it is up to the manager to decide how much influence others are given, this style can only work well, when both managers and staff understand and are in agreement about which tasks are important. The staffs expertise, experience and intuition need to be encouraged, not stifled, if challenging situations are to be negotiated. The goal of the participative leader is to persuade followers to share their values and connect with their vision. It appears to me, to be obvious that most large organizations, the federal government, the military, etc., require leaders and followers that possess the same core Leadership Styles :: essays research papers The style of leadership that would be the most comfortable for me would be delegation. One of the first signs of good supervision is effective delegation. Delegation is when supervisors give responsibility and authority to subordinates to complete a task, and let the subordinates figure out how the task can be accomplished. Effective delegation develops people who are ultimately more fulfilled and productive. Managers become more fulfilled and productive themselves as they learn to count on their staffs and are freed up to attend to more strategic issues. Delegating is a critical skill for supervisors. Delegating involves working with an employee to establish goals, granting them sufficient authority and responsibility to achieve the goals, often giving them substantial freedom in deciding how the goals will be achieved, remaining available as a resource to help them achieve the goals, assessing their performance, addressing performance issues and/or rewarding their performance. Ultimately, the supervisor retains responsibility for the attainment of the goals, but chooses to achieve the goals by delegating to someone else. True delegation means giving up a little of what we would like to hold onto while keeping what we might prefer to give up. Participative style of leadership would be least comfortable for me. A participative leader, rather than making decisions, looks to involve other staff in the process. Often however, as it is up to the manager to decide how much influence others are given, this style can only work well, when both managers and staff understand and are in agreement about which tasks are important. The staffs expertise, experience and intuition need to be encouraged, not stifled, if challenging situations are to be negotiated. The goal of the participative leader is to persuade followers to share their values and connect with their vision. It appears to me, to be obvious that most large organizations, the federal government, the military, etc., require leaders and followers that possess the same core

Analysis of Fire and Ice by Robert Forst Essay

This article tries to analyze the unique features in structure, words, phonology, syntax and rhetoric in the poem of 40-Love by British poet Roger McGough in order to have a deeper understanding of the content and form of a poem. Keywords: McGough, 40-Love, Love, Style 1. Introduction It is well known that in a variety of literary genres, the form of poetry has been stressed most. Efforts have been exerted on the skillful combination of rhythm and structure to create numerous great works all over the world. Modern American poet E. E. Cummings (1884-1962) is a good case in point. He is famous for odd style, novel and unique form in the poetic world. His â€Å"l (a† has been regarded as the â€Å"the most elegant and beautiful structure of the literature created by Cummings†. (Kennedy, 1980). Therefore, his poems are renowned as â€Å"poem picture† or â€Å"visual poetry†, or the concrete poetry that we are quite familiar with. The features of it is that vivid visual images of words coming from irregular syllables, letters, punctuation, syntax, format and print strengthen the internal imagination of poetry, deepen the artistic conception, convey and enrich the connotation. (Abrams, 2005). Coincidentally, besides E. E. Cummings, contemporary British poet Roger McGough (1937- ) is another master in writing concrete poems. His 40-Love can be considered as one of the greatest concrete poems. McGough is the second of the three in Liverpool Group. The other two are Henry Adrian Henri and Patan Brian Patten. McGough, born in 1937, 5 years younger than Henry, is always in naughty mentality. His poems are full of secular fun and display more profound life from the perspective of a child. This article, from the viewpoint of stylistics, analyzes the features of structure, words, phonology, syntax and rhetoric in the poem of â€Å"40-Love† in order to gain a eeper understanding of this poem. 2. Stylistic Analysis Greek poet Simonides once said, â€Å"Poetry is the picture with sound while the pictures are the silent poems. † (Zhu, 2005). That is to say, the content of a poem must be combined with its form to achieve its perfection, namely, the combination of form and spirit, what we often cheris h. Here we will try to find how Mcgough do it in his â€Å"40-Love†. The poem tells that a middle-aged couple is playing tennis. Then they go home. But the net is still between them. It reflects the gap between middle-aged couples. I will quote the poem here to help to explain my opinion. 40-Love (Peng, 2000) middle couple tenwhen game and go the 118 aged playing nis the ends they home net Asian Social Science will be tween 2. 1 Structure still be – them June, 2009 As a whole, the poem has a total of 20 words. But the two words â€Å"tennis† and â€Å"between† are separated by hyphens to be symmetric in structure. The words in the poem are set in two sequences, like two sides of the couple. The middle blank or empty is like a net to separate the two. There are only two words in each line to symbolize the bouts of the ball. The title of 40-love, the top of the net, is right on the top of the poem, signifying the scoreboard. This poem looks like a tennis court with a net being used to separate the words. It is like a tennis game. This side serves and the other side hits back. Many bouts form the poem. The invisible net is like the barrier between the middle-aged couple. Even if they finish the tennis game, they still have the net, which still exists invisibly. However, it is this net that they can depend on to handle their marriage and have the responsibilities not to break the rules. There is a net in tennis and there are rules to obey. With the net, there are more difficulty and more interests. So, accordingly, more training and attention is a must. Imagining that, when playing tennis with no net or rules, people would feel difficult to last their games for longer period. In addition, only the letter of â€Å"L† of â€Å"Love† in the title is capitalized and the rest is de-capitalized, which shows that, to some extent, the couple has not been in the pursuit of the perfect details again, because love between the middle-aged couple has faded away. Furthermore, there is no punctuation in the whole poem, indicating that life of marriage is closed and uninteresting. Since there is no end, gap appears. 2. 2 Words The poet pays special attention to the words in the poem. First of all, the title â€Å"40-Love† one of interests embodied in the poem. The figure of 40 stands for the age of middle-aged people. And 40-Love is a scoring term in tennis. Tennis scoring is love, 15, 30 and 45 in sequence. Love here means zero. Three goals scores 40. No goals, no score. Thus, the title is of pun with two meanings. One refers to be 40-year-old love and the other is 3:0. Whether 40-year-old love is vain or not depends on attitudes of the two parties. Let come to two words of â€Å"middle† and â€Å"aged† in the first line. â€Å"Middle-aged† means people are in their midlife. The poet deliberately separates it to achieve the reunification of form and others. It also symbolizes that middle-aged husband and wife can not be integrated again. The two important words of â€Å"tennis† and â€Å"between† are placed in two vertical columns to get a metaphorical meaning that there is an invisible net in the emotional world of the man and the woman. They are not intimate any longer. Game† in the fifth line can be referred as either play game or sport. The scoring in tennis competition is more complex. Tennis game has games and sets. In a game, those who win 15, 30 and 45 will get one point. And the player who gets 6 points will win one set. In the poem, the couple does not finish even one game and go home since they hav e a deep estrangement. â€Å"Still† in the ninth line shows that the middle-aged husband and wife have ineffable anguish and can not get rid of their unpleasantness and gain relaxed though they make concessions as far as possible. 2. Phonology Words in the poem are basically monosyllabic. They are mechanical and boring to read and easy to suggest that the life of the couple is dull and lack of amenities. From the perspective of phonology, the short vowel such as /i/, is used for many times in the poem to leave the pressing impression to the readers to realize the urgent emotional crisis of the middle-aged couple. But there is slowness in the urgency. The diphthong /ei/ and / u/ are employed to slow down the speech rate and demonstrate that the middle-aged people have become calm and unhurried when dealing with things. Especially, the long vowel / i: / in â€Å"tween† in the last line leaves enough time for the middle-aged couple in crisis to think over the issue. In addition, there are rhymes in the poem, such as, middle and couple, ten and when, game and they, go and home, will and still. Rhymes here give the readers boredom, and symbolize the dull life of the middle-aged couple. Moreover, the alliterations of be and be-, tween and them, make the two words close and imply that the middle-aged husband and wife still have the ties that can not be cut off although there is a gap between them. 2. Syntax For the convenience of analysis, I rearrange the order of the whole poem: middle aged couple playing ten-nis when the game ends and they go home the net will still be be-tween them. First of all, from the angle of tense, the plain and flat present tense, used from the beginning to the right end of the poem, indicates the dull or prosaic marriage life of the middle-aged couple. Nevertheless, â⠂¬Å"playing† is used unconventionally. If â€Å"plays† is used here, readers will know that the couple play tennis often rather than occasionally. There will be not much gap between them. Playing† indicates that there is absence of regular communication between the husband and wife. It stresses that it is just at this moment that they are playing tennis. In sentence structure, there is a time adverb â€Å"when† to combine the sentence. As usual, however, there is no conjunction of â€Å"and† between the main clause and the subordinate clause. Thus â€Å"and† is added here to deliberately create a loose state, suggesting that there is no close contact between the husband and wife. And there should be an adversative conjunction of â€Å"but† in front of the next sentence â€Å"the net will still be be-tween them†. As we 119 Vol. 5, No. 6 Asian Social Science all know, an adversative conjunction word usually give people an unexpected, rising and falling impression. The word â€Å"but† is omitted here to inevitably imply that life of the middle-aged couple is no longer full of ups and downs, great happiness or sadness. 2. 5 figure of speech Poets often use figure of speech because, as Emily Dickinson once said, the mission of a poet is to â€Å"speak the truth, but to put it in an implicit way† in order to seize the readers’ interest and stimulate their imagination. In the poem of â€Å"40-love†, the poet employs the figure of speech, e. g. metaphor. On the one hand, in form, the blank along the net is like a net to suggest the gap between the middle-aged couple. On the other hand, everyday life is like playing games. Everybody hit the ball to the others. Such routine game results in no passion at last. Moreover, in my opinion, the writing technique of understatement is employed in this poem. The tone of the whole poem is calm, without any fluctuating. However, it is the deliberate understatement that discloses the theme of the poem incisively and vividly. . Conclusion This poem written by McGouph with unique style has rich connotation in its unique form. In this poem with perfect combination of the spirit and form, the emotional gap of the middle-aged couple can be discerned and expressed by the stylistic techniques in the structure, words, phonology, syntax and figure of speech. In Mending Wall, a poem written by American poet Robert Fr ost, the neighbor is intransigence and stubborn. Even at the last line of the poem, he still murmurs that â€Å"good fences make good neighbors† (GU, 2005). Every couple, therefore, especially the middle-aged couples, should pull the fence between them down, believing â€Å"good communication makes good couples†. References Abrams, M H. (2005). A Glossary of Literary Terms. Beijing: Foreign language Teaching and Reasearch Press. Gu, Zhengkun. (2005). Treasury of Appreciating English Poems, Volume of Poetry. Kennedy, Richard S. (1980). Dreams in the Mirror: A Biography of EE Cummings. New York: Liveright. Peng, Yu. (2000). Two Concrete Poems. College English. Zhu, Guangqian. (2005). Poetics. Shanghai Century Publishing Group. 120

Background information on how the development LASIK The WritePass Journal

Background information on how the development LASIK Introduction Background information on how the development LASIK IntroductionBackground information on how the development LASIKThe way the LASIK has affected peoples’ lives;ConclusionRelated Introduction Over the thousands last years, human has experienced many ways to redress their sight. One of the most significant inventions was an eyeglass which is discovered in 1268 by Roger Bacon. This invention has developed through creation contact lenses (Teagle Optometry, 2007). However, the majority of humans who wear glasses or contact lenses are bothering from it. Therefore, Jose Barraquer discovered LASIK in 1950, which is considered one of the recent technologies in vision correction. LASIK is a Latin word that indicates to create  a thin layer  of the cornea  (black  eye)  and then  using the laser  vision correction (LASIK Portal, 2010). Background information on how the development LASIK Lasik is one of the most important types of eye surgery in advance medical that intended for improving in particular Myopia, Hyperopia and Astigmatism. It was invention by Jose Barraquer at the first time, where he effectuated the first operation     to reduce the  thin  flaps  in the cornea  to change its shape by keratomileusis. By 1981, the Alaximr Laser was been founded which worked on ultraviolet radiation, and it was used at the first time by Rangaswamy Srinivasan to decrease tissue in specific way   through extract layers of thin films without any effects thermal in surrounding area. As a result, he could use this kind of Laser without any side effect compared to different type of Laser which worked in the field of visible radiation. After a number of experiments, the Lasik technique has been improved in 1990 by Ioannis Pallikaris and Lucio Buratto to become more accuracy than keratomileusis. All of these results led Stephen Brint and Stephen Slade to performed s urgery operation in the United States for the first time (Ezine Articles, 2011). With the development of technology, Lasik has become more fast than before and it has been improved to be better (Wikipedia, 2011). The way the LASIK has affected peoples’ lives; . The majority of humans bother from wearing spectacles or contact lenses therefore they want to eliminate them by Lasik. Lasik has many positive effects on humans including that Lasik has ability to accurately correct most layers of Myopia, Hyperopia and Astigmatism. Moreover, its surgery occupies five to ten minutes with painless or very little pain. It is one of the easiest operations because it is operated by computer and does not require any stitches after it.  Ã‚   One of the most important factors of Lasik that most patients are not longer needed corrective glasses. Conclusion

Life After High School by Joyce Carol Oates Essays

Life After High School by Joyce Carol Oates Essays Life After High School by Joyce Carol Oates Paper Life After High School by Joyce Carol Oates Paper Essay Topic: High School In Joyce Carol Oates’s short story, Life After High School, the character’s wear masks to fit into the late 1950’s strict society that accompany them throughout the story. As the other character’s masks begin to unravel; Zach is fixated on living a normal life that his mask inevitably ends his existence. While Zach’s mental instability ends his life, his obsession saves Sunny and Tobias by removing their veils before it was too late. The main character Zachary Graff, a typical awkward teenager, excels greatly in school, but Zach’s intelligence masks his mental instability. He falls in love with the perfect, ideal, girl in high school. In reality, Zachary loves his best friend Tobias, but the constraints of the 1950’s judgmental society led him to believe that Sunny would be the perfect choice to portray a heterosexual character. He lived a conflicted life up until his death, after being rejected by both Sunny and Tobias, he felt as though death was his only way to freedom. Zach owns a 1956 Plymouth which is envied by many; this represents the masculinity that Zachary lacks. The irony is seen when the car becomes Zach’s coffin instead of expressing his sexuality. Zach expresses little to no interest in girls other than Sunny. His classmates remember him as almost antisocial, some even called him a loner. Zachary shunned sports but claims to have a liking to golf which suggests that even thoug h he lacked talent in the sport, it was accepted by his father. Barbara â€Å"Sunny† Bushman, known as the popular, â€Å"too good to be true† Christian. She represents the perfect, All-American 1950’s girl. Sunny can tell that Zach has become infatuated with her by the way he lingers around the school a little too long just to drive her home. Sunny, a devout Christian, flattered by Zachary’s gestures, unfortunately, knows that she can not tell Zachary to get lost. It just simply is not in her vocabulary after being given the ni

Disproportionate Minority Contact Essays

Disproportionate Minority Contact Essays Disproportionate Minority Contact Essay Disproportionate Minority Contact Essay Jake Huston 11116850 Criminal Justice 205 10/30/11 Research on police and prosecutors reveals that uniformly they disagree that discrimination occurs in their agency and office. What then explains the disproportionate minority contact that occurs and the disparate treatment within the prosecutor’s office? Although police and prosecutors may contend that discrimination does not occur within their agency but that does not mean discrimination doesn’t occur. The facts show that minorities are targeted much more than whites. There are many factors that contribute to this. I don’t believe it is any one agency that specifically targets minorities but rather the criminal justice system as a whole. The interplay between the media, the criminal justice system, and the public has a huge influence on discrimination within the system. Another big factor in the disproportionate minority contact is the fact that the poor, troubled inner cities are filled with mostly minorities. The culture within these poor neighborhoods perpetuates a crime mentality that becomes an easy target for the criminal justice system. I argue that the discrimination doesn’t come from a racial bias but rather from the criminal culture created in poor areas. The media has a lot of power over the criminal justice system. The media’s ability to change public perception is a key element in the discrimination we see in the police and prosecutors agencies. As the media plays up an issue the general public responds with fear. This in turn puts pressure on police to crack down on the issue. The police widen the net and arrest more offenders for lesser crimes. This makes it appear that crime goes up due to the increased number of arrests. The media communicates this to the public and people become even more concerned. This causes legislators to make changes in the laws such as mandatory minimums. We are all exposed to the discrimination portrayed in the media. Most people describe the typical criminal as a young black male. Most people think this way because it is what we see being covered on the news. In the video Law and Disorder in Philadelphia the policemen said they are able to tell right away who is a criminal and who isn’t. This is blatant discrimination. The video also shows that the police are concentrated in poorest parts of Philadelphia that have the most crime. These projects are filled with minorities. If the police are concentrated in the poor areas with minorities and not in the predominantly white suburbs there will obviously be more minorities being arrested than whites. The culture of the inner city greatly helps to perpetuate crime. Thomas Winston in the documentary Life and Death of a Dealer talked about how growing up he felt that there was no option besides crime. He started selling drugs at the age of 13. He also said that a dealer can make $15,000 a week but working minimum wage only yields about $110. (1) In the book Code of the Street by Elijah Anderson he describes how the culture in the streets is accepting of drug trafficking. On page 110, subchapter THE CULTURAL ECONOMIC CONNECTION, Anderson says that the lack of jobs has made the underground economy an easy and lucrative industry to enter. He talks about how a family whose main concern is paying bills wont let the criminality deter them. If you can’t find a job you are going to find some way to make money. This acceptance of criminality creates many problems. In Law and Disorder in Philadelphia the cop says that he couldn’t point out one house in the neighborhood that isn’t involved in the heroin trade. This is possible because of the culture in very poor area. (2) Anderson refers to the book the Philadelphia Negro written by W. E. B. Du Bois in 1899. Du Bois said that the â€Å"submerged tenth† is characterized by irresponsibility, drinking, violence, robbery, thievery, and alienation. He also said that the problem that kept young African American men from jobs is a lack of education, connections, social skills, and white skin color. These are all true today, over 100 years later. People in these neighborhoods don’t trust the police and generally refuse to help them in any way. People here don’t live by the same code of ethics that the rest of society does. (3) Anderson says the â€Å"code of the street† is a set of informal rules governing impersonal public behavior, particularly violence. This is evident in the first chapter as he describes a trip down Germantown Avenue in Philadelphia. At the top of the street is the upper class area people stroll down the street openly with no fear. It is racially diverse where you see blacks, whites, and all other races mixing socially. The buildings are all very well maintained. As you move down the street into the poor area the buildings start to see bars and the windows, they look rundown and some have even collapsed. It becomes much less racially diverse and the street corners and open areas are filled with mostly young blacks. The middle and upper class blacks from up the street do not associate themselves with the lower class. 4) On pg. 50 Anderson tells about how he overheard a black person mutter â€Å"street nigger† to a black friend after they had a small altercation with a low class black. This shows that the middle and upper class does not associate themselves with the poor and that there is a distinct cultural difference. Everyone here has a much different attitude. Instead of the carefree strolls up the stree t you see everyone â€Å"watching their back†. There is a flagrant disregard for laws in this area. (4) Anderson talked about seeing a young teen walking through Vernon Park drinking a beer in broad daylight. This is something I would never expect to see in a park near my home. After reading Code of the Street and watching both videos I can easily see why there is the disproportionate minority contact in the criminal justice system. The poor inner city is filled with crime due to lack of opportunity. People are going to do whatever they have to do to survive. Since the inner cities are filled with minorities they are the ones targeted by police. If the inner city were filled with white people, there would be the same criminal culture.

Aircraft Maintenance Planning Procedures Research Paper

Aircraft Maintenance Planning Procedures - Research Paper Example It also contains maintenance control and Flight Crew advisory information that is used during routine operations between schedule maintenance to main base. Purpose of a technical log is for recording malfunctions and defects discovered during operations and for recording details of maintenance and information relevant to flight safety (Phil, 120) The technical log can only be certified by Civil Aviation Authority. b. Data recording Flight data recorder (FDR) maintenance is found fitted in the aircraft. This is for purposes of investigating an accident as a priority amongst other measures. Aircraft operators use the FDR for quantifying maintenance action that is needed. This is by confirming reported operation of the aircraft and its systems. The FDR performs function checks, operation checks, reasonableness, quality, drop out, data download, data conversions, parameters, simulations, analogue and digital data, engineering units, and stimulation. All these are necessary for an airwort hy aircraft. It is highly recommended for all aircraft data to be recorded electronically. They should be recorded on a daily basis to ensure aircraft airworthiness. These records are then kept as part of maintenance records for a particular aircraft. An automatic generation of records has been adopted in aircrafts (Thomas, 7). c. Maintenance schedule This contains details of what is required for maintenance of an aircraft and when it should be done. The maintenance schedule is created by the publisher i.e. the Original Equipment Manufacturer (OEM) or the Type Certificate Holder of the Aircraft. The CAA has to approve the product once the Aircraft Engineers have done a thorough checkup of the product airworthiness (Dinesh, 201). The maintenance checks involved here are airframe, engine, propeller and other equipment check. The maintenance manual is kept in the aircraft pertaining each equipment and parts of the aircraft. Under CAP 411, an aircraft should not exceed 2730kg for light aircrafts under the Light Aircraft Maintenance Schedule (LAMS). A maintenance schedule is compiled by first reviewing the manuals prepared by the OEM. Intervals are usually analysed based on the flying hours, flight cycles or the calendar time. A combination of these factors can also be used. The tasks are usually combined depending on the approved intervals. In order to compile work packages, tasks can be done earlier that recommended; this is a general rule which is often used. The CAA can however, allow for a later date, if such an agreement is reached or in exceptional cases. It is important to note that tasks which have more than one frequency should be given preference to govern what occurs first. The frequency is however, affected by the aircraft intended operations. TCH provides maintenance planning documents for tasks to be undertaken in large aircrafts. Aircrafts typical flight profile should be matched with the aircraft type. For example; two flying hours to each flight c ycle, seven flying hours to each flight cycle and so on. We can deduce that maintenance for the first aircraft is maintained more often than the second aircraft in our example. Schedules should thus be developed for the particular type of operation. Also considerations for maintenance frequency are made in regards to area of

Globalization and Identity Essay Example | Topics and Well Written Essays - 2250 words

Globalization and Identity - Essay Example However, it escalated to the period of colonial expansion due to the potential resources, which are abundant in the lands of Asia (Steger 29). Almost all parts of Asia have been subjugated from colonial rule due to such a heated demand to sustain the needs of industrialized nations in terms of raw material supply (Scupin 325). Most Western countries have colonized the Asian lands. India was controlled by Britain. The Philippines by Spain then the United States. The first point I want to analyze, in a critical sense, is the politics, economics and social conditions that surround identity; specifically, the people of a particular nation. One cannot undermine that certain countries in Asia have governments, whether it is an imperial or a feudal one. China, for example, has an imperial government that is centralized. It cannot be denied that China, under political circumstances, already has a sense of belonging and expansion of influence. Other countries, for the same matter, have feudal societies and tribes that already have a systematized government, possibly ethnic or tribal to a certain extent. Identification is present. These established governments are already propagating a sense of identity. The intervention came from the Western countries since they are forwarding a certain political ideology on their part. This would only mean that the West tries to manipulate the identity of the people in these countries to ensure that they are adher ent to the conventions, which are in favor of the Western people. ... Mostly Western countries have colonized the Asian lands. India was controlled by Britain. Philippines by Spain then United States. The first point I want to analyze, in a critical sense, is the politics, economics and social conditions that surround identity; specifically, the people of a particular nation. One cannot undermine that certain countries in Asia have governments, whether it is an imperial or a feudal one. China, for example, has an imperial government that is centralized (Scupin 325). It cannot be denied that China, under political circumstances, already has a sense of belonging and expansion of influence (Steger 24). Other countries, for the same matter, have feudal societies and tribes that already have a systematized government, possibly ethnic or tribal to a certain extent (Nye 162). Identification is present. These established governments are already propagating a sense of identity. Intervention came from the Western countries since they are forwarding a certain pol itical ideology on their part (Nye 163). This would only mean that the West tries to manipulate the identity of the people in these countries to ensure that they are adherent to the conventions, which are in favour of the Western people. One cannot deny that governments have changed and wars between the colonizers and the colonized burst out due to an assertion of independence and self – governance. In terms of economics, identity is indeed affected. Trading happened between countries of the West and Asia. Cultural exchange is one of the crucial things that must be considered (Steger 24). Upon the exchange of goods and technology, one cannot neglect that there will be

Native people social movements Essay Example | Topics and Well Written Essays - 1000 words

Native people social movements - Essay Example that was formed, the Anishinabe continued to fight following the established prophecy hence finding themselves in California and being united helped the Anishinabe to obtain back their nationality from the whites. Afterwards, Ojibwa also known as the Anishinabe people’s urge with their friends resulted in a fight that begun killing one another for the purpose of the hunting ground. All the Anishinabe people with their tribes became vigilant in protecting their people from slavery and their territories (Lorman). However, they all fought for the protection of their homeland for the land became the main issue that brought all the suffering. With the native youth movements, roadblocks were put in place to stop invasion into their land since they abolished all of the following activities. Railways construction, highways, mining, resorts, dams, cities, deep seaports, garbage dams and many others that led to their outbreak of war. Additionally, different organizations movements were created like the American-Indian political activism during the year of 1960s for obtaining their rights (Bruchac). In addition, among other movements was a national association for the advancement of colored people (NAACP), southern Christian leadership conference (SCLC) and finally groups were also formed like National organization for women (NOW). The formed group and movements mainly dealt with the rights of their individuals together with land issues maintaining their social integrity. During the establishment of national Indian youth council (NICY) that occurred after the tribes of Oklahoma with Great Plains that defeated the NCAI of 1994. The developed groups used peaceful ways with Americans where they encouraged the third world liberations. However, various groups of young individuals came up with American Indian movements (AIM) with an intention of the police harassment (Williamson). However, the Alcatraz Island reduces the pride and their consciousness with the rise of

Engineering Managment Coursework Example | Topics and Well Written Essays - 1500 words

Engineering Managment - Coursework Example The company was founded by John Cadbury in the year 1824. It is presently headquartered at Uxbridge, London, United Kingdom and operates in more than 50 countries of the world (Prinz, 2012). Since the time of its inception, the company has been growing at a rapid pace and the reason behind its continued success is its clear strategies, plans, goals and objectives. The company has been operating in the market for almost 200 years. Despite economic crises that have affected the world from time to time, the company is still standing strong in the market. Moreover, the company has always also remained focused on its mission, objectives and strategies. However, the company has faced some kind of issues in the form of PR crisis and financial crisis. To deal with this, it is highly recommended that the company should reduce its operating cost and appoint a dedicated public relations officer to oversee any public relations issue. Question 1 1.1 Cadbury’s current mission, objectives, a nd strategies Mission The core purpose of the company is to work together and create a brand which people love. The core mission of the company is to reach the world and become a leading company in the confectionary industry. Cadbury wants to be a part of people's lives through their products. In addition, the company’s mission is to eradicate poverty in the areas of its operation and create a work environment which promotes work force diversity. Objectives The objectives of a company generally depend upon the business situations. Similarly, the business objectives of Cadbury have changed over the years according to the market situations. Cadbury has always relied on short term goals, rather than the long term objectives. However, their short term goals have changed on an average of 10 years. The objectives of the company are as follows: - To open a Cadbury shop and increase its profitability. To use new technology to increase the production. The company has been able to fulf il most of its objectives till now and the current objective of the company is to maintain the level it has already reached. To provide high return to the shareholders. To achieve revenue and sales growth. Strategies The strategy of the company to achieve its goals is very much straight forward. The business strategy of the company is to tap new markets for its products and increase its profitability. Nevertheless, since most of the products of the company are sold all over the world, it has formulated a two-pronged growth strategy, which is dependent upon the cash flow of the company. In addition, the company is also looking forward to new channels of product distribution in order to increase sales. Since, 70 % of the total sales of the company come from impulse purchase that is why the company is also targeting restaurants, pubs and petrol stations to sell their products. Apart from the mission statements, strategies and objective the organizational culture of

Administration Essay Example | Topics and Well Written Essays - 500 words

Administration - Essay Example Likewise, knowledge, change and globalization are the three driving force of this era. Consequently, various transitions were brought out; and so most businesses go all-out to adapt to this phenomenon.First, most businesses today were highly globalized, they were conducted by the used of modern gadgets. People from around the world are working together to achieve a common goal. Even they were being raised with different cultures and beliefs, their mutual interests bind them to do business together. To think about all this happenings, the role of the administration is quiet hard, isn't it Back in the 90's, businesses conducted communication through the use of fax machines or telephones. This time, two individuals from two countries can do a conversation without any communication barriers-face to face. However, consequent to this are the technical, cultural and linguistic challenges of globalizations hence managers have to adjust in the midst of these diversities. Technically, as globa lization takes its place, old business entities need to update all their gadgets to compete in the global market or take advantage of the rapid technological change.

Engineering Managment Coursework Example | Topics and Well Written Essays - 1500 words

Engineering Managment - Coursework Example The company was founded by John Cadbury in the year 1824. It is presently headquartered at Uxbridge, London, United Kingdom and operates in more than 50 countries of the world (Prinz, 2012). Since the time of its inception, the company has been growing at a rapid pace and the reason behind its continued success is its clear strategies, plans, goals and objectives. The company has been operating in the market for almost 200 years. Despite economic crises that have affected the world from time to time, the company is still standing strong in the market. Moreover, the company has always also remained focused on its mission, objectives and strategies. However, the company has faced some kind of issues in the form of PR crisis and financial crisis. To deal with this, it is highly recommended that the company should reduce its operating cost and appoint a dedicated public relations officer to oversee any public relations issue. Question 1 1.1 Cadbury’s current mission, objectives, a nd strategies Mission The core purpose of the company is to work together and create a brand which people love. The core mission of the company is to reach the world and become a leading company in the confectionary industry. Cadbury wants to be a part of people's lives through their products. In addition, the company’s mission is to eradicate poverty in the areas of its operation and create a work environment which promotes work force diversity. Objectives The objectives of a company generally depend upon the business situations. Similarly, the business objectives of Cadbury have changed over the years according to the market situations. Cadbury has always relied on short term goals, rather than the long term objectives. However, their short term goals have changed on an average of 10 years. The objectives of the company are as follows: - To open a Cadbury shop and increase its profitability. To use new technology to increase the production. The company has been able to fulf il most of its objectives till now and the current objective of the company is to maintain the level it has already reached. To provide high return to the shareholders. To achieve revenue and sales growth. Strategies The strategy of the company to achieve its goals is very much straight forward. The business strategy of the company is to tap new markets for its products and increase its profitability. Nevertheless, since most of the products of the company are sold all over the world, it has formulated a two-pronged growth strategy, which is dependent upon the cash flow of the company. In addition, the company is also looking forward to new channels of product distribution in order to increase sales. Since, 70 % of the total sales of the company come from impulse purchase that is why the company is also targeting restaurants, pubs and petrol stations to sell their products. Apart from the mission statements, strategies and objective the organizational culture of

Fermentation Kinetics of Different Sugars Essay

Fermentation Kinetics of Different Sugars - Essay Example The findings achieved through the experiment showed an increased rate of fermentation in tubes with glucose and sucrose as the substrate while lactose showed a massively decreased rate of fermentation. The addition of sodium fluoride also caused a decreased rate of fermentation. Analysis of the complete data suggested that the carbohydrates used by Saccharomyces cerevisiae for fermentation play a great role in the final rate of fermentation. Keywords: Saccharomyces cerevisiae, fermentation, carbohydrates, magnesium Fermentation Cells and tissues irrespective of belonging to animal or plant have a minimum requirement of energy. Different processes such as synthesis of molecules, transportation, DNA replication and cell repairs have varying requirements of energy. To successfully complete these processes cells undertake many metabolic processes to achieve their supply of energy. Glucose being the most important carbohydrate and the end product of almost all food sources is the beginnin g point of these metabolic processes. Energy conversion starts from the process of glycolysis. As explained by Agrimi et al., (2011) glycolysis begins with the entry of a single glucose molecule and terminates with the production of two pyruvate molecules. The process immediately yields four ATP molecules. However, with the consumption of two ATP molecules at two different steps in the cycle, the net production via substrate level phosphorylation turns out to be two. Although the process itself is not affected by the presence or absence of oxygen, the final production of the ATPs is hugely affected under hypoxic conditions as only 2 ATP molecules per glucose are produced instead of 36 ATP molecules per every glucose molecule. Depending on the availability of oxygen the pyruvates produced at the end of glycolysis are either shuttled into either cellular respiration / Krebs cycle or they are used in the process of fermentation. Fermentation has been derived from a Latin word ‘fe ver’ meaning to ferment.

Analysis of Conan Doyles work Essay Example for Free

Analysis of Conan Doyles work Essay English coursework : A comparative essay on 2 Sherlock Holmes stories  The world has chosen to remember Sir Arthur Conan Doyle chiefly for his creation of the fictional master detective, Sherlock Holmes. This prestigious character has been hugely popular for over one hundred years shown in many different ways, whether it be books, television series, magazine articles, and so on.  Conan Doyle himself was born in Edinburgh, rather than the London setting that Sherlock Holmes lives and works. He actually set out to be an oculist, however when no patients came he had plenty of time to write his stories. Around the same time, The Strand magazine was first published, and Sherlock Holmes was printed for the first time in its pages. His stories were not long enough to be books of their own, and thrived as a regular part of their magazine. The Sherlock Holmes stories are written in a very upper class setting. Watson and Holmes take cabs everywhere, and have very high class mannerisms and habits, such as leaving a calling card if the person they have visited is not there. And Holmes himself carries round a cane with him a rather posh acquirement thought to be that of a gentleman. They also have a resident in the home to look after them both, and take care of the household, which could of course only be afforded by those of the higher class.  The fact that these stories were written in such a way is easily explained. The stories were written for the magazine, The Strand. In those times, very little people read magazines, and could afford to subscribe. Most people read newspapers, but these magazines were aimed at the higher class, and particularly for the gentleman, because a very small number of women were expected to be able to read, and so they did not benefit from such a publication. The Speckled Band and, The Red-Headed League are both very interesting stories. The structures of both are much the same, but that accounts for all of Conan Doyles creations in accordance to the Sherlock Holmes stories.  In accordance to this structure, The Speckled Band begins with a visit to the house from someone needing Holmess help. An element included right at the beginning of the story is the deduction Holmes makes of Helen. This is extremely typical of Conan Doyle as it is a feature he nearly always adds as each of his stories unfold. In The Speckled Band Holmes instantly deduces that she travelled early by dog cart on heavy roads to the station before travelling by train to London. The next lines describe Helen giving a violent start and staring at Holmes in bewilderment. This is also underlyingly very typical of Conan Doyle to include such a description, as it what he includes in all of his stories at this point. In direct comparison, The Red Headed League features a swift deduction of Jabez Wilson in which he concludes that the man done at some point manual labour, takes snuff, has a freemason, has been in China, and has done a considerable amount of writing recently. The line directly following on from that is set out exactly the same as in The Speckled Band and describes Jabez to Start up in his chair, with his forefinger upon the paper, but his eyes upon Holmes So we can conclude from this that this is an element Conan Doyle likes to include in his stories which also suitably gives a first time reader an idea of the way Holmess mind works.  Following on from that, a long monologue is heard which tells every detail of the story. This is unusual in stories written now. We can note that in real life no one tells a story in such detail, uninterrupted, and this reflects tracts of today. But it is actually very typical of writings at the time.

Vedic Mathematics Multiplication

Vedic Mathematics Multiplication Abstract Vedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual in its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method. Introduction Is Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered. Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West. The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous. If that doesn’t sound magical enough, its interesting to note that the word ‘Vedic’ means coming from ‘Vedas’ a Sanskrit word meaning ‘divinely revealed.’ The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as ‘Om’ are said to have been revealed by the Heavens themselves. According to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic. Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable. The Vedic system also provides for the solution of difficult problems in parts; they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole system of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy. The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one ‘accurate’ method. This leads to more creative, fascinated and intelligent pupils. Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc. But the real beauty and success of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible. Now having known that even the 16 sutras are the Jagadguru Sankaracharya’s invention we mention the name of the sutras and the sub sutras or corollaries in this paper. The First Sutra: EkÄ dhikena PÃ…Â «rvena The relevant Sutra reads EkÄ dhikena PÃ…Â «rvena which rendered into English simply says By one more than the previous one. Its application and modus operandi are as follows. (1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method. A. First method B. Second Method This is the whole working. And the modus operandi is explained below. Modus operandi chart is as follows: (i) We put down 1 as the right-hand most digit 1 (ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit. (iii) We multiply that 2 by 2 and put 4 down as the next previous digit. (iv) We multiply that 4 by 2 and put it down thus 8 4 2 1 (v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 Ãâ€" 2 = 138 and so on). (vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step. (vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand. (viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there. Our chart now reads as follows: The Second Sutra: Nikhilam Navataņºcaramam Daņºatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navataņºcaramam Daņºatah, which literally translated means: all from 9 and the last from 10. We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication. Suppose we have to multiply 9 by 7. 1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power. Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin); 3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign (–) between them, to show that the numbers to be multiplied are both of them less than 10. 4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts. 5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 – 10) i.e. 6 as the left hand part of the answer 9 + 7 – 10 = 6 The First Corollary The first corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency. This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clear: Suppose one wants to square 9, the following are the successive stages in our mental working. (i) We would take up the nearest power of 10, i.e. 10 itself as our base. (ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 12 (iv) Thus 92 = 81 The Second Corollary The second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context. Its literal meaning is the same as before (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the previous one is 1. So one more than that is 2. Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 Ãâ€" 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 Ãâ€" 2 / 25 = 2 / 25. Now we proceed on to give the third corollary. The Third Corollary Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows: i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product. The following example will make it clear: The Third Sutra: Ã…Â ªrdhva TiryagbhyÄ m Ã…Â ªrdhva TiryagbhyÄ m sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number. The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold. A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13. (i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer; (ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and (iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 Ãâ€" 13 = 156. The Fourth Sutra: ParÄ vartya Yojayet The term ParÄ vartya Yojayet which means Transpose and Apply. Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 Ãâ€" –2 = –2x2 but we have 7x2 in the divident. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x As for the third term we already have –2 Ãâ€" 9x = –18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient. Q = x2 + 9x + 24 Now the last term of the quotient multiplied by – 2 gives us – 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder ∠´ Q = x2 + 9x + 24 and R = 53. The Fifth Sutra: SÃ…Â «nyam Samyasamuccaye Samuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time. Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator. Fourthly Samuccaya means combination or total. Fifth meaning: With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations. Sixth meaning With the same sense (total of the word Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero. Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time. The Sixth Sutra: Äâ‚ ¬nurÃ…Â «pye Ã…Å ¡Ãƒâ€¦Ã‚ «nyamanyat As said by Dani [32] we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in {pp. 51, 74, 249 and 286 of [51]}. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean proportionately and the first by the first and the last by the last. Here the later subsutra acquires a new and beautiful double application and significance. It works out as follows: i. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 2x2 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 : 4 and the ratio of the second part to the last coefficient i.e. 1 : 2 are the same. Now this ratio i.e. x + 2 is one factor. ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thus Thus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used. The Seventh Sutra: Sankalana VyavakalanÄ bhyÄ m Sankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples. A concrete example will elucidate the process. Suppose we have to find the HCF (Highest Common factor) of (x2 + 7x + 6) and x2 – 5x – 6 x2 + 7x + 6 = (x + 1) (x + 6) and x2 – 5x – 6 = (x + 1) ( x – 6) the HCF is x + 1 but where the sutra is deployed is not clear. The Eight Sutra: PuranÄ puranÄ bhyÄ m PuranÄ puranÄ bhyÄ m means by the completion or not completion of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic and forth degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section. The Ninth Sutra: CalanÄ  kalanÄ bhyÄ m The term (CalanÄ  kalanÄ bhyÄ m) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: YÄ vadÃ…Â «nam YÄ vadÃ…Â «nam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah. The Eleventh Sutra: Vyastisamastih Sutra Vyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing. The Twelfth Sutra: Ã…Å ¡esÄ nyankena Caramena The sutra Ã…Å ¡esÄ nyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57! Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few trivial examples for the same. The Thirteen Sutra: Sopantyadvayamantyam The sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra. The illustration given by them. The proof of this is as follows. The General Algebraic Proof is as follows. Let d be the common difference Canceling the factors A (A + d) of the denominators and d of the numerators: It is a pity that all samples given by the book form a special pattern. The Fourteenth Sutra: EkanyÃ…Â «nena PÃ…Â «rvena The EkanyÃ…Â «nena PÃ…Â «rvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows. For instance 43 Ãâ€" 9. i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product The Fifthteen Sutra: Gunitasamuccayah Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors. Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 6x2 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz. Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient; i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6. This is a very simple and easy but absolutely certain and effective process. The Sixteen Sutra :Gunakasamuccayah. It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product. In symbols we may put this principle as follows: Sc of the product = Product of the Sc (in factors). For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80. Similarly in the case of cubics, biquadratics etc. the same rule holds good. For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 Ãâ€" 3 Ãâ€" 4 = 1 + 6 + 11 + 6 = 24. Thus if and when some factors are known this rule helps us to fill in the gaps. Literature Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation. Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas. The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them: We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2). Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division. Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it. This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol à ª5à ª, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000; Presmeg, 1986; van Hiele, 2002). Such visualisation accesses the brains holistic activity (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure. The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998; Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research considered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions. Methodology The research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boy’s and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school). Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation. The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised. The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial expression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered. Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer. Results The first question (1a) in each questionnaire was a two-digit multiplication. In the first, it was 37 Ãâ€" 58, and the second 23 Ãâ€" 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 Ãâ€" 69: 2 times 9 is 18 3 times 9 + 2 times 6 is 39 + carried 1 = 40 3 times 6 + carried 4 = 22 Expansion of binomials A summary of the results in the first of the algebra questions (Q2 see Figure Vedic Mathematics Multiplication Vedic Mathematics Multiplication Abstract Vedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual in its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method. Introduction Is Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered. Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West. The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous. If that doesn’t sound magical enough, its interesting to note that the word ‘Vedic’ means coming from ‘Vedas’ a Sanskrit word meaning ‘divinely revealed.’ The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as ‘Om’ are said to have been revealed by the Heavens themselves. According to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic. Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable. The Vedic system also provides for the solution of difficult problems in parts; they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole system of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy. The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one ‘accurate’ method. This leads to more creative, fascinated and intelligent pupils. Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc. But the real beauty and success of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible. Now having known that even the 16 sutras are the Jagadguru Sankaracharya’s invention we mention the name of the sutras and the sub sutras or corollaries in this paper. The First Sutra: EkÄ dhikena PÃ…Â «rvena The relevant Sutra reads EkÄ dhikena PÃ…Â «rvena which rendered into English simply says By one more than the previous one. Its application and modus operandi are as follows. (1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method. A. First method B. Second Method This is the whole working. And the modus operandi is explained below. Modus operandi chart is as follows: (i) We put down 1 as the right-hand most digit 1 (ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit. (iii) We multiply that 2 by 2 and put 4 down as the next previous digit. (iv) We multiply that 4 by 2 and put it down thus 8 4 2 1 (v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 Ãâ€" 2 = 138 and so on). (vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step. (vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand. (viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there. Our chart now reads as follows: The Second Sutra: Nikhilam Navataņºcaramam Daņºatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navataņºcaramam Daņºatah, which literally translated means: all from 9 and the last from 10. We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication. Suppose we have to multiply 9 by 7. 1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power. Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin); 3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign (–) between them, to show that the numbers to be multiplied are both of them less than 10. 4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts. 5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 – 10) i.e. 6 as the left hand part of the answer 9 + 7 – 10 = 6 The First Corollary The first corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency. This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clear: Suppose one wants to square 9, the following are the successive stages in our mental working. (i) We would take up the nearest power of 10, i.e. 10 itself as our base. (ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 12 (iv) Thus 92 = 81 The Second Corollary The second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context. Its literal meaning is the same as before (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the previous one is 1. So one more than that is 2. Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 Ãâ€" 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 Ãâ€" 2 / 25 = 2 / 25. Now we proceed on to give the third corollary. The Third Corollary Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows: i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product. The following example will make it clear: The Third Sutra: Ã…Â ªrdhva TiryagbhyÄ m Ã…Â ªrdhva TiryagbhyÄ m sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number. The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold. A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13. (i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer; (ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and (iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 Ãâ€" 13 = 156. The Fourth Sutra: ParÄ vartya Yojayet The term ParÄ vartya Yojayet which means Transpose and Apply. Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 Ãâ€" –2 = –2x2 but we have 7x2 in the divident. This means that we have to get 9x2 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x As for the third term we already have –2 Ãâ€" 9x = –18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient. Q = x2 + 9x + 24 Now the last term of the quotient multiplied by – 2 gives us – 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder ∠´ Q = x2 + 9x + 24 and R = 53. The Fifth Sutra: SÃ…Â «nyam Samyasamuccaye Samuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time. Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator. Fourthly Samuccaya means combination or total. Fifth meaning: With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations. Sixth meaning With the same sense (total of the word Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero. Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time. The Sixth Sutra: Äâ‚ ¬nurÃ…Â «pye Ã…Å ¡Ãƒâ€¦Ã‚ «nyamanyat As said by Dani [32] we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in {pp. 51, 74, 249 and 286 of [51]}. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean proportionately and the first by the first and the last by the last. Here the later subsutra acquires a new and beautiful double application and significance. It works out as follows: i. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 2x2 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 : 4 and the ratio of the second part to the last coefficient i.e. 1 : 2 are the same. Now this ratio i.e. x + 2 is one factor. ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thus Thus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used. The Seventh Sutra: Sankalana VyavakalanÄ bhyÄ m Sankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples. A concrete example will elucidate the process. Suppose we have to find the HCF (Highest Common factor) of (x2 + 7x + 6) and x2 – 5x – 6 x2 + 7x + 6 = (x + 1) (x + 6) and x2 – 5x – 6 = (x + 1) ( x – 6) the HCF is x + 1 but where the sutra is deployed is not clear. The Eight Sutra: PuranÄ puranÄ bhyÄ m PuranÄ puranÄ bhyÄ m means by the completion or not completion of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic and forth degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section. The Ninth Sutra: CalanÄ  kalanÄ bhyÄ m The term (CalanÄ  kalanÄ bhyÄ m) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: YÄ vadÃ…Â «nam YÄ vadÃ…Â «nam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah. The Eleventh Sutra: Vyastisamastih Sutra Vyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing. The Twelfth Sutra: Ã…Å ¡esÄ nyankena Caramena The sutra Ã…Å ¡esÄ nyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57! Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few trivial examples for the same. The Thirteen Sutra: Sopantyadvayamantyam The sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra. The illustration given by them. The proof of this is as follows. The General Algebraic Proof is as follows. Let d be the common difference Canceling the factors A (A + d) of the denominators and d of the numerators: It is a pity that all samples given by the book form a special pattern. The Fourteenth Sutra: EkanyÃ…Â «nena PÃ…Â «rvena The EkanyÃ…Â «nena PÃ…Â «rvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows. For instance 43 Ãâ€" 9. i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product The Fifthteen Sutra: Gunitasamuccayah Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors. Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 6x2 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz. Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient; i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6. This is a very simple and easy but absolutely certain and effective process. The Sixteen Sutra :Gunakasamuccayah. It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product. In symbols we may put this principle as follows: Sc of the product = Product of the Sc (in factors). For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80. Similarly in the case of cubics, biquadratics etc. the same rule holds good. For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 Ãâ€" 3 Ãâ€" 4 = 1 + 6 + 11 + 6 = 24. Thus if and when some factors are known this rule helps us to fill in the gaps. Literature Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation. Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas. The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them: We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2). Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division. Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it. This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol à ª5à ª, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000; Presmeg, 1986; van Hiele, 2002). Such visualisation accesses the brains holistic activity (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure. The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998; Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research considered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions. Methodology The research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boy’s and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school). Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation. The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised. The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial expression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered. Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer. Results The first question (1a) in each questionnaire was a two-digit multiplication. In the first, it was 37 Ãâ€" 58, and the second 23 Ãâ€" 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 Ãâ€" 69: 2 times 9 is 18 3 times 9 + 2 times 6 is 39 + carried 1 = 40 3 times 6 + carried 4 = 22 Expansion of binomials A summary of the results in the first of the algebra questions (Q2 see Figure